I remember facing my second legendary hound in Pokémon gold, know that they'd run away/growl the first move and my first pokémon couldn't prevent it so I instantly tossed one of Kurt's balls and actually managed to catch it instantly, best childhood memory. I always wondered the % to actually manage to catch Suicune with a fast ball from full health.

the most amazing thing ive ever done was walking into the cave with mewtwo in it and forgetting i only had one weak ghastly and just said what the heck and through a ultra ball at it and caught it while it had full health nothing else that day could compare to the happiness i felt when that happened
if you read this i appreciate it and wish you all a good day

Legendaries are assholes I actually used a masterball on a zubat once just for the hell of it and it broke free and ran away. Yet I caught Raikou with a timer ball while he was sleeping. I think pokemon just likes to **** with you.

There' s always room for failures, that was one of them.
It's never happened since then as you said the chances are probably less than one.. I guess around 0.75% of missing since it is the best ball in the game.

It's actually not an amazing once-in-a-life-time thing, even with a .75% chance, that's still only a 1/133 chance, which if you think about it, you have less of a chance of rolling quads ( a 1/1000 chance) than missing a catch with a masterball.

**onederman rolls 1,719**
There are not 1000 numbers in 0-999 (*roll 4*) , there are 10,000 numbers because of 1 through 9999 plus the 0 equals ten thousand or 10000, and a 1 in 10 chance of quads, so it's 10/10000 = 1/1000, not 10/1000 = 1/100. Rolling ones (1, 2, 3, etc.) is 100% chance, rolling dubs ( 00, 11, 22, 33, etc.) is 10% chance, rolling trips (000, 1111, 222, 333, etc.) is 1% chance, so rolling quads (I think you get the point) is .1% or 1/1000 chance.
Wow, that was a lot more than I meant to type.